Find the function with the given derivative whose graph passes through the point P.
-$$g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 4,4 )$$
A) $g ( x ) = \frac { 1 } { x } + x ^ { 2 } + \frac { 49 } { 4 }$
B) $g ( x ) = \frac { 1 } { x } + x ^ { 2 } - \frac { 49 } { 4 }$
C) $g ( x ) = - \frac { 1 } { x } - x ^ { 2 } - \frac { 49 } { 4 }$
D) $g ( x ) = - \frac { 1 } { x } + x ^ { 2 } - \frac { 49 } { 4 }$

Question

Quizplus · Accepted Answer

Integrating the given derivative, we get:
$$g(x)=\int \left(\frac{1}{x^2}+2x\right)dx=-\frac{1}{x}+x^2+C$$
To find C, we use the point given that the graph passes through P(-4,4). Substituting x=-4 and g(x)=4 in the above equation, we get:
$$4=-\frac{1}{-4}+(-4)^2+C$$
Simplifying and solving for C, we get C=-49/4.
Therefore, the function is: 
$$g(x)=-\frac{1}{x}+x^2-\frac{49}{4}$$

Find the Function with the Given Derivative Whose Graph Passes g′(x)=1x2+2x,P(−4,4)g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 4,4 )g′(x)=x21​+2x,P(−4,4)

Find the Function with the Given Derivative Whose Graph Passes $g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 4,4 )$